Kinetic theory of collisionless self-gravitating gases. II. Relativistic corrections in galactic dynamics

In this paper we study the kinetic theory of many-particle astrophysical systems imposing axial symmetry and extending our previous analysis in Phys. Rev. D 83, 123007 (2011). Starting from a Newtonian model describing a collisionless self-gravitating gas, we develop a framework to include systematically the first general relativistic corrections to the matter distribution and gravitational potentials for general stationary systems. Then, we use our method to obtain particular solutions for the case of the Morgan and Morgan disks. The models obtained are fully analytical and correspond to the post-Newtonian generalizations of classical ones. We explore some properties of the models in order to estimate the importance of post-Newtonian corrections and we find that, contrary to the expectations, the main modifications appear far from the galaxy cores. As a by-product of this investigation we derive the corrected version of the tensor virial theorem. For stationary systems we recover the same result as in the Newtonian theory. However, for time dependent backgrounds we find that there is an extra piece that contributes to the variation of the inertia tensor.

Figure 1

Rotation curves for the 1PN version of the second MM member for different values of the parameter $\lambda$: (a) $\lambda ={10}^{-3}$, (b) $\lambda =5\times {10}^{-3}$, (c) $\lambda =2\times {10}^{-2}$, and (d) $\lambda =7\times {10}^{-2}$. For $\lambda \sim {10}^{-2}$ or more, the differences with the Newtonian model (dashed line) become significant. In this case, the rotation curve tends to flatten as $\lambda$ grows.

Figure 2

Mass profiles for the 1PN version of the second MM member for the same values of the figure 1. As expected, the differences with the Newtonian model (dashed line) become significant for $\lambda \sim {10}^{-2}$ or more. In the case (d), we have a profile in which the maximum is not at $R=0$.